# About a question of Gateva-Ivanova and Cameron on square-free set-theoretic solutions of the Yang-Baxter equation

## TL;DR

This paper introduces a new sequence to better estimate the size of minimal involutive square-free solutions to the Yang-Baxter equation, improving previous bounds and providing new counterexamples to a longstanding conjecture.

## Contribution

It proposes a novel sequence for estimating solution sizes and constructs counterexamples to Gateva-Ivanova's Conjecture, advancing understanding of set-theoretic solutions.

## Key findings

- Improved bounds for the cardinality of minimal solutions
- Construction of new counterexamples to Gateva-Ivanova's Conjecture
- Enhanced estimation methods for square-free solutions

## Abstract

In this paper, we introduce a new sequence $\bar{N}_m$ to find a new estimation of the cardinality $N_m$ of the minimal involutive square-free solution of level $m$. As an application, using the first values of $\bar{N}_m$, we improve the estimations of $N_m$ obtained by Gateva-Ivanova and Cameron and by Lebed and Vendramin. Following the approach of the first part, in the last section we construct several new counterexamples to the Gateva-Ivanova's Conjecture.